First Order Circuit

| First Order Circuits |

A first order circuits can only contain one energy storage element ( a capacitor or and inductor). The circuit will also contain resistance. So there are two types of First Order Circuits.

> RC circuit

> RL circuit

| Source-Free RC Circuits |

A source-free circuit is one where all independent sources have been disconnected from the circuit after some switch action. The voltages and currents in the circuit typically will have some transient response due to initial conditions (initial capacitor voltages and initial inductor currents). We will begin by analyzing source-free circuits as they are the simplest type. Later we will analyze
circuits that also contain sources after the initial switch action.

Consider the RC circuit shown below. Note that it is source-free because no sources are connected to the circuit for t > 0. Use KCL to find the differential equation:

| Checks on the solution |

> Verify that the initial condition is  satisfied.

> Show that the energy dissipated over all time by the resistor equals the initial energy stored in the capacitor.

General form of the D.E. and the response for a 1st-order source-free circuit

In general, a first-order D.E. has the form:

Solving this differential equation (as we did with the RC circuit) yields:

where 

τ = (Greek letter “Tau”) = time constant (in seconds)

| Source-Free RL Circuit |

Consider the RL circuit shown below. Use KCL to find the differential equation:

>Equivalent Resistance seen by an Inductor

>For the RL circuit in the previous example, it was determined that τ = L/R.

As with the RC circuit, the value of R should actually be the equivalent (or Thevenin) 

resistance seen by the inductor.

>In general, a first-order RL circuit has the following time constant:

| My Learning Experience XD |

An RLC circuit (the letters R, L and C can be in other orders) is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively. The circuit forms a harmonic oscillator for current and will resonate in a similar way as an LC circuit will. The main difference that the presence of the resistor makes is that any oscillation induced in the circuit will die away over time if it is not kept going by a source. This effect of the resistor is called damping.